This has led to several methods that provide inner approximations for the PSD cone, some of which may still lead to the same SDP solution. Examples include, the set of diagonally dominant (\(DD_n\) ) matrices and scaled diagonally dominant (\(SDD_n\) ) matrices, which can be written as conic combinations of rank one PSD matrices and conic combinations of second order cones, respectively [1]}, and lead to approximations that can be solved using
linear programs (LP) or resp. second-order cone programs (SOCP).
Along the same line, Factor-width \(k\) matrices, which can be viewed as a generalization of scaled diagonally dominant matrices [2]}, can be decomposed as a combination of PSD matrices of rank at most \(k\) . Finally, one could exploit chordal sparsity patterns of matrices [3]}, i.e., sparsity induced by a chordal graph, to equivalently express a large semidefinite constraint as a set of smaller semidefinite constraints. Unfortunately, the scalability of chordal decompositions is limited by the size of the largest clique.
Moreover, approximating the PSD cone with \( DD_n\)  or \( SDD_n\)  is conservative, which means that either feasible solutions of the constrained problem may be sub-optimal or that no solution exists. Further, enforcing these constraints is problematic as \(n\)  increases, since the number of constraints exhibit a combinatorial growth [1]}, [5]}.
